A110682 A convolution triangle of numbers based on A027307.

1, 2, 1, 10, 4, 1, 66, 24, 6, 1, 498, 172, 42, 8, 1, 4066, 1360, 326, 64, 10, 1, 34970, 11444, 2706, 536, 90, 12, 1, 312066, 100520, 23526, 4672, 810, 120, 14, 1, 2862562, 911068, 211546, 42024, 7410, 1156, 154, 16, 1

Offset

0,2

Comments

Triangle T(n,k) for A(x)^k = Sum_{n>=k} T(n,k)*x^n, where o.g.f. A(x) satisfies A(x) = (1+x*A(x)^2)/(1-x*A(x)^2). - Vladimir Kruchinin, Mar 16 2011

Links

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Formula

T(0, 0) = 1; T(n, k) = 0 if k<0 or if k>n; T(n, k) = Sum_{j, j>=0} T(n-1, k-1+j)*A006318(j).

Sum_{k, k>=0} T(n, k) = A108442(n+1).

T(n,k) = k/(2*n-k)*Sum_{i=0,n-k} binomial(2*n-k,n-k-i)*binomial(2*n-k+i-1,2*n-k-1), n >= k > 0. - Vladimir Kruchinin, Mar 16 2011

Mathematica

T[n_, k_] := (k/(2*n - k))*Sum[Binomial[2*n - k, n - k - j]*Binomial[2*n - k + j - 1, 2*n - k - 1], {j, 0, n - k}]; Table[T[n, k], {n, 0, 25}, {k, 1, n}] // Flatten (* G. C. Greubel, Sep 05 2017 *)

Prog

(PARI) for(n=0, 25, for(k=1, n, print1((k/(2*n-k))*sum(i=0, n-k, binomial(2*n-k, n-k-i)*binomial(2*n-k+i-1, 2*n-k-1)), ", "))) \\ G. C. Greubel, Sep 05 2017

Crossrefs

Columns: A027307, A032349, A033296.

Sequence in context: A142963 A099755 A202483 * A110327 A105615 A136216

Adjacent sequences: A110679 A110680 A110681 * A110683 A110684 A110685

Keyword

nonn,tabl

Author

Philippe Deléham, Sep 15 2005

Status

approved